Cross-temporal forecast reconciliation: Optimal combination method and heuristic alternatives

Forecast reconciliation is a post-forecasting process aimed to improve the quality of the base forecasts for a system of hierarchical/grouped time series. Cross-sectional and temporal hierarchies have been considered in the literature, but generally, these two features have not been fully considered together. The paper presents two new results by adopting a notation that simultaneously deals with both forecast reconciliation dimensions. (i) The closed-form expression of the optimal (in the least squares sense) point forecasts fulfilling both contemporaneous and temporal constraints. (ii) An iterative procedure that produces cross-temporally reconciled forecasts by alternating forecast reconciliation along one single dimension (either cross-sectional or temporal) at each iteration step. The feasibility of the proposed procedures, along with first evaluations of their performance as compared to the most performing ‘single dimension’ (either cross-sectional or temporal) forecast reconciliation procedures, is studied through a forecasting experiment on the 95 quarterly time series of the Australian Gross Domestic Product from Income and Expenditure sides. For this dataset, the new procedures, in addition to providing fully coherent forecasts in both cross-sectional and temporal dimensions, improve the forecast accuracy of the state-of-the-art point forecast reconciliation techniques.